Metamathematics, machines, and Godel's proof

How much computers can be used to prove mathematical theorems is a question of great interest. Some would claims that Gödel's incompleteness theorem means that there are severe limits on what a computer can do. Well, Metamathematics, machines and Gödel's proof presents a computerized proof of Gödel's incompleteness theorem itself. The author uses the lisp programming language, firstly to write a program to check proofs for the first order set-theoretic language Z2, and secondly to express the proof checking program itself in Z2, using the Boyer-Moore theorem prover to prove the existence of an unprovable sentence

I wouldn't, however, recommend this book to the novice - its more for those with some experience of computer-based proofs, or possibly for those interested in what can be done with the lisp language. Even for those familiar with arithmetical proofs, the connection with the programs in the book isn't immediately clear, especially as it uses Z2 rather than Peano Arithmetic. But the more advanced approach does result in a more reliable form of proof than might otherwise be the case - a true computer based proof, rather than just using a computer to help persuade someone of the truth of a theorem.

The automatic verification of large parts of mathematics has been an aim of many mathematicians from Leibniz to Hilbert. While GÃ¶del's first incompleteness theorem showed that no computer program could automatically prove certain true theorems in mathematics, the advent of electronic computers and sophisticated software means in practice there are many quite effective systems for automated reasoning that can be used for checking mathematical proofs. This book describes the use of a computer program to check the proofs of several celebrated theorems in metamathematics including those of GÃ¶del and Church-Rosser. The computer verification using the Boyer-Moore theorem prover yields precise and rigorous proofs of these difficult theorems. It also demonstrates the range and power of automated proof checking technology. The mechanization of metamathematics itself has important implications for automated reasoning, because metatheorems can be applied as labor-saving devices to simplify proof construction.